However, if \(A\)  is not closed, or if \(\mu \)  is of infinite energy, there is in general no measure \(\nu \)  which would be uniquely determined within \(\mathfrak {M}^+_A\)  by the equality \(\kappa _\alpha \nu =\kappa _\alpha \mu \)  n.e. on \(A\)  (see e.g. Remark REF ). Nevertheless, a substantial theory of inner \(\alpha \) -Riesz balayage of arbitrary \(\mu \in \mathfrak {M}^+\)  to arbitrary \(A\subset \mathbb {R}^n\)  was developed [1]}–[2]}, and this was performed by means of several alternative approaches described below.The outer \(\alpha \) -Riesz balayage was investigated by J. Bliedtner and W. Hansen [3]} in the general framework of balayage spaces. See also N.S. Landkof [4]}, where however it was imposed upon \(A\)  and \(\mu \)  certain restrictions, e.g. that \(A\subset \mathbb {R}^n\)  be Borel while \(\mu \in \mathfrak {M}^+\)  be bounded.
